The letter to Weil included a number
of striking conjectures which eventually
changed much of the direction of
research in automorphic forms. Some of their consequences were
explained in a graduate course given at Princeton
in the spring of 1967, and then things were put in
a somewhat wider context in a series of lectures at Yale
later that Spring. These notes were previously published
as the first of the Yale Mathematical Monographs.

This monograph was based on lectures given early
in April of 1967 at Yale University, thus
several months after the letter to Weil.
Nonetheless it is reticent about the conjectures
formulated in that letter. Results are
formulated in terms of the dual group introduced there,
which could for the groups of the lectures be introduced
without any reference to the Galois group because only split
groups are treated. There is, however, only the slightest of allusions
to any generalization of class-field theory: the observations that what can be done for one
reductive group should be done for all and that
the identification of an automorphic
L-function with an Artin L-function or with a Hasse-Weil L-function is tantamount to
a reciprocity law.
These two observations underline that
functoriality arose in
an attempt to find a nonabelian class-field theory under the
influence of the view, which arose in the early sixties, that much of the
theory of automorphic forms could and should be treated in the context of
group representations. The major technical impulse was the need for a concisely defined
general class of Euler products that included those arising from the theory of Eisenstein series.

The formula (6), which is established in sufficient generality to verify
the convergence of the Euler products, is not established in general, although it is
surmised that it is generally true. This was, indeed, proved a little later in
complete generality and, so far as I know, quite independently by
Ian MacDonald (Spherical functions on a group of p-adic type).
I had heard
of his result, even though his monograph was not yet available, by the time
Problems in the theory of automorphic forms
was written, so that I could simply
invoke it. The formula now carries, quite rightly, his name.

The formula referred to as the formula of Gindikin-Karpelevich was, indeed, proved
in general by them, but had first been discovered by Bhanu-Murty and proved by him
for the special linear group over R in

Although the notes
for the lectures were available as a preprint at the time they were
delivered or shortly thereafter, the monograph did not appear until 1970. Apart from
the addition of one or two footnotes and the correction of misprints and slips of the pen,
there were no alterations.

Gelbart and Shahidi have written a useful survey
of the theory of automorphic L-functions,
Analytic properties of automorphic L-functions. I recommend it to the reader of
Euler products.

Letter to Godement

The letter to Weil that saw the birth
of the L-group was written in January, 1967.
Somewhat later that same year, Roger Godement asked Langlands to comment on
the Ph. D. thesis of Hervé Jacquet. His reply included a number
of conjectures on Whittaker functions for both real and
p-adic reductive groups. These were later to be proven,
first in the p-adic case by Shintani for GLn and Casselman & Shalika
in general,
and much later in the real case by a longer succession
of people.

This letter, a report on Jacquet's
thesis, is undated, but a letter from Godement dated May 12th, 1967 asks
that the report be submitted before the end of May. I assume it was sent from
Princeton so as to arrive in Paris before the date requested.

The notation may cause the reader some difficulties.
Some symbols, for example ,
have meanings that change (sometimes explicitly
but sometimes only implicitly) in the course of the letter.
There is a particularly dangerous lapse in regard to .
Other symbols, sometimes the same,
are employed in ways that have become uncommon. The symbol
appears, for example, as a representation of a compact group. The notation
< a, > for the
value of the multiplicative function
at the group element a is particularly disconcerting.

References to pages either in Jacquet's thesis or in the handwritten
letter have been allowed to stand.

The formula for Whittaker functions for
unramified representations suggested
in the letter was proved by Casselman and Shalika.

It appears from the Institute records that
Godement visited Princeton early in
March of 1967. It must have been then that
I spoke to him. The lectures at
Yale were given early in April of 1967 and appeared later as
the monograph Euler Products
(included just above).

Eisenstein series (the book)

This was written in 1965 and distributed for many
years in a famous purple mimeographed
document by the Yale University Mathematics
Department. It was later published in the Springer Lecture Notes
series (volume #544).

This originally appeared as a
supplement to an article by A. Borel and H. Jacquet
in Automorphic forms, representations, and L functions,
Proceedings of Symposia in Pure MathematicsXXXIII,
A. M. S., 1979.